1. Multiple Regression Analysis
y = b0 + b1x1 + b2x bkxk + u
3. Asymptotic Properties
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Consistency
Under the Gauss-Markov assumptions OLS is BLUE, but in other cases it won’t always be possible to find unbiased estimators
In those cases, we may settle for estimators that are consistent, meaning as n  ∞, the distribution of the estimator collapses to the parameter value
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3. Sampling Distributions as n 
n1 < n2 < n3 n2 n1 b1
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Consistency of OLS
Under the Gauss-Markov assumptions, the OLS estimator is consistent (and unbiased)
Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness
Will need to take probability limit (plim) to establish consistency
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Proving Consistency
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A Weaker Assumption
For unbiasedness, we assumed a zero conditional mean – E(u|x1, x2,…,xk) = 0
For consistency, we can have the weaker assumption of zero mean and zero correlation – E(u) = 0 and Cov(xj,u) = 0, for j = 1, 2, …, k
Without this assumption, OLS will be biased and inconsistent!
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7. Deriving the Inconsistency
Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case
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8. Asymptotic Bias (cont)
So, thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted variable
Main difference is that asymptotic bias uses the population variance and covariance, while bias uses the sample counterparts
Remember, inconsistency is a large sample problem – it doesn’t go away as add data
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9. Large Sample Inference
Recall that under the CLM assumptions, the sampling distributions are normal, so we could derive t and F distributions for testing
This exact normality was due to assuming the population error distribution was normal
This assumption of normal errors implied that the distribution of y, given the x’s, was normal as well
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10. Large Sample Inference (cont)
Easy to come up with examples for which this exact normality assumption will fail
Any clearly skewed variable, like wages, arrests, savings, etc. can’t be normal, since a normal distribution is symmetric
Normality assumption not needed to conclude OLS is BLUE, only for inference
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Central Limit Theorem
Based on the central limit theorem, we can show that OLS estimators are asymptotically normal
Asymptotic Normality implies that P(Z<z)F(z) as n , or P(Z<z)  F(z)
The central limit theorem states that the standardized average of any population with mean m and variance s2 is asymptotically ~N(0,1), or
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Asymptotic Normality
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13. Asymptotic Normality (cont)
Because the t distribution approaches the normal distribution for large df, we can also say that
Note that while we no longer need to assume normality with a large sample, we do still need homoskedasticity
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Asymptotic Standard Errors
If u is not normally distributed, we sometimes will refer to the standard error as an asymptotic standard error, since
So, we can expect standard errors to shrink at a rate proportional to the inverse of √n
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15. Lagrange Multiplier statistic
Once we are using large samples and relying on asymptotic normality for inference, we can use more that t and F stats
The Lagrange multiplier or LM statistic is an alternative for testing multiple exclusion restrictions
Because the LM statistic uses an auxiliary regression it’s sometimes called an nR2 stat
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LM Statistic (cont)
Suppose we have a standard model, y = b0 + b1x1 + b2x bkxk + u and our null hypothesis is
H0: bk-q+1 = 0, ... , bk = 0
First, we just run the restricted model
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LM Statistic (cont)
With a large sample, the result from an F test and from an LM test should be similar
Unlike the F test and t test for one exclusion, the LM test and F test will not be identical
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18. Asymptotic Efficiency
Estimators besides OLS will be consistent
However, under the Gauss-Markov assumptions, the OLS estimators will have the smallest asymptotic variances
We say that OLS is asymptotically efficient
Important to remember our assumptions though, if not homoskedastic, not true
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